Episode 21: Doubts and Uncertainty

Ken was in the lab early, having fled the lunchtime robotics conversation. One day of trying to wrangle mechs into shape without even knowing what that shape should be had been more than enough for him. Even Kelly Jean, who usually managed to make herself unwelcome, had joined the group eagerly awaiting the B-league team lists. Ken pulled his notebook out of his bag and dropped it onto the bench with more clatter than was strictly necessary. He flipped through to the notes Mathematician Dustborn had given him for his modelling project.

Apparently modelling an atom was more complicated than anyone thought a first year student could do on his own. Ken had insisted that he was willing to put in extra work to figure it out, but Mathematician von Rejk had told him that he needed to simplify if he wanted to be able to finish and sent him to Dustborn. Mathematician Dustborn had suggested that he model just a single electron in a molecule of the chemical benzene. Ken sort of understood why this was simpler. Most of the electrons in the molecule interacted with all the other pieces in a complicated way — which was why modelling an atom with lots of pieces would be difficult — but the electron he was modelling just sort of floated around the outside, for reasons he didn’t fully understand. Looking at the equations he was trying to show made sense, Ken admitted to himself that he was relieved von Rejk had made him take a simpler route. There were derivatives everywhere and solving equations with derivatives was much harder than the regular kind. Especially when they were partial derivatives instead of the regular kind, which was another thing Mathematician Dustborn had explained quickly and Ken didn’t really understand. He figured that at least it would give him something to talk about in his write up at the end of the project.

“You’re early, Ken.” Bryony arrived in the lab and dropped her things on the lab bench across from him.

“I’m escaping the masochistic roboticists,” he said and surprised himself at how grumpy he sounded. “What about you?”

“Being late is bad for my complexion.” Bryony sat down. “I don’t think they’re masochistic, though. Robo-duelling is a legit hobby, although I absolutely empathise with the feeling of having heard way too much about it in the last couple of weeks.”

“Yeah, I dunno what they see in it. I think it’s a waste of time.” Ken shrugged. “Not that anyone’s listening to me. How’s the project going?”

“It’s pretty fun. Von Rejk let me extend something I was doing on my own with a sort of new litmus-type test — it’s not really new, but I’m doing it a bit differently to normal — and it definitely helps to write everything down. Not just lab notebook type stuff, although that’s important too, obviously, but what you expect to happen and why and how you’re going to test it. You can’t test an hypothesis you haven’t formulated yet!”

“So you’re doing lots of experiments and stuff?”

“Not really, actually. I mean, I want to do lots of experiments and stuff, but realising that somebody’s actually going to look at how I do this and mark me on whether or not I’m actually using the scientific method kind of slows me down. And I’m trying to figure out how to incorporate some of the ideas from the last project, about what you’re actually measuring when you do tests. Because you’re never really exact, are you? There’s always some amount you could be out by.”

“It’s a bit disconcerting, really. I think I prefer maths, where a number is a number. In measurements it seems like a number is just an indication of more or less where you are.”

“Hey, it’s kind of like in analysis class. The reading is in a neighbourhood of the actual answer.”

It took Ken a moment to remember exactly how they’d defined a neighbourhood, but Bryony’s description did have some sense to it. He was still trying to puzzle it out when Mathematician Nieminen began the session.

“Good afternoon, all of you. I know you’re eager to get to work on your project for Mathematician von Rejk, but first I’m going to ask you to briefly cast your minds back to your previous project, where we were looking at measurements. I have all your write-ups, with comments, here.” She tapped the lecturer’s desk at the front of the lab.

“I’ll be giving them back for you to look over in a minute,but first I want to discuss a topic that all of you have come at from one or another angle in your reports: uncertainty. You’ve all realised that if you measure, say, the width of a pin with both a metre stick and a micrometer, you won’t get exactly the same result. All you’ll be able to read off the metre stick is something like ‘a millimetre’, while the micrometer will take you into decimal places of millimetres. But if they’re both correctly calibrated we want the readings to be consistent. So how do we resolve that?”

Ken looked pointedly at Bryony in the silence that followed, but she shook her head and refused to answer. He gave her a few seconds to change her mind and said, “To quote an anonymous classmate, they’re both in some neighbourhood of the right value.”

“Oh very good,” Nieminen said. “I like that. There might be some debate about whether or not the right value can even exist in principle, but the idea of getting an interval is absolutely right. Whose idea was that, by the way?” She looked around the class.

“It was me, but I didn’t think it actually meant anything,” Bryony said. “I’m not exactly sure what Ken means, to be honest.”

“Hmm, well then.” Nieminen turned to sketch a number line on the screen at the front of the lab. “If we read ‘one millimetre’ off the metre stick, we might be tempted to say that the reading is just this.” She placed a dot at one on the numberline. “But really, there’s some uncertainty in the reading. I might be reading half a millimetre too much or too little, so what I really get is more like this.” She drew a bracket along the number line starting at one half and stretching past the original dot to one-and-a-half. “I can write that as one plus or minus a half.” She added to the screen, below the bracket, 1±0.5.

“Now my micrometre reading is also going to have some uncertainty attached, so it’ll be a different bracket. The uncertainty should be much smaller, of course. If I read one point two millimetres off the micrometer, I can put a bracket here.” She drew a tiny bracket in. “We won’t worry about the exact value of the uncertainty for now.

“Since the brackets overlap, our readings are consistent. On the other hand, if the micrometer measured three-” She stepped down the number line and drew a tiny bracket in at three. “We wouldn’t overlap at all. Hopefully you’d agree intuitively that something would be wrong.”

Several people nodded.

“Explicitly stating our uncertainties allows us to quantify that discomfort. Now, what if the reading I take from the micrometre is just a smidgen more than one point five?” Nieminen drew a bracket around the end of the big metre stick bracket.

Ken stared at the screen thoughtfully. It was much more satisfying to think of the measurements as being completely overlapping. You expected the micrometer to improve on the reading from the metrestick. But the brackets did overlap for this case, even if it was just a little.

“I guess it’s fine,” Buhle said. “As long as the real value is in the part where they overlap. I mean, I’m not sure if you can tell that from the measurements, but they must be basically consistent if that’s an option. It gets kind of complicated, though.”

Mathematician Nieminen laughed. “It does indeed get complicated, but we’ll develop the ideas as we go. We’ll do some exercises later on, but for now all I’m expecting is that you report your measurements as intervals, like this.” She gestured to the 1&pm;0.5. “Talk to us about how to get it right for your particular situation. If you start to feel like we’re glossing over things, that’s good, because we are. We’ll have more sessions like these in the future, especially as you gain some of the mathematical tools that are helpful for dealing with the ideas. Now I’ll let you get to work — oh wait, John?” She turned to look at Mathematician Dustborn, who was standing in the doorway.

“Just a quick announcement, Monica. We’ve got seats for the class at the opening robo-duelling match this weekend. I’ll drop all the paperwork in your common room. Please make sure I get it back by tomorrow morning. Miss Adams, as we discussed previously, you will be excused. Now, back to work with you.” He whisked out of the room to a buzz of chatter.

Bryony looked like she was trying not to smile, although Ken wasn’t sure why. She’d been supportive enough of the class’s obsession. “How’d you manage to get out of it then?”

“No, it’s not like that.” She looked up and down the table. “Ricardo banned me from the first class excursion and I’m just relieved it’s not all that disappointing. Not that I wouldn’t have enjoyed watching the match.”

“That sounds like it comes with a story,” Ken said. “But if you got out of it, maybe I can too.”